# What is the difference between Lorentz transformations and proper time

1. Jul 2, 2014

### albertrichardf

Hi all,
What is the difference between Lorentz transformations and yt?. That is, the Lorentz transformations for moving between two reference frames are not the same as the relativistic ones.

For example considering a frame F that is stationary and an inertial frame F' with velocity v. Time dilation for frame F' is given by yt, where gamma is the relativistic factor. But the Lorentz transformations give y(t - [xv/c2 ]). These are obviously different. So which one is correct? Or is time dilation a special case of a transformation? And Length contraction is also not the same as space transformations. So is it a special case as well?

2. Jul 2, 2014

### ghwellsjr

Time Dilation is based on the speed of an object in an Inertial Reference Frame. If you know its speed, you know its Time Dilation factor, gamma.

If you define a worldline for an object in one IRF and mark events defining increments of Proper Time along that worldline based on its instantaneous speed and then you transform to any other frame moving inertially with respect to the first IRF, those events will automatically be adjusted for the correct Proper Time in the new IRF.

3. Jul 2, 2014

### bcrowell

Staff Emeritus
The Lorentz transformation can't be reduced to length contraction and time dilation. If that was all it was, then a Lorentz transformation would just be equivalent to a rescaling of units, e.g., changing units from seconds and light-seconds to years and light-years.

Nor are time dilation and length contraction possible special cases of the Lorentz transformation. There is no value of v for which a Lorentz transfromation looks like a pure time dilation, a pure length contraction, or just a combination of the two.

Time dilation is a description of what happens in a certain complicated measurement process, e.g., one in which you act out the twin paradox and each twin carries a clock. Length contraction is a description of what happens to measurement with a ruler when the ruler is not at rest relative to the object being measured, and the ends of the ruler are matched up with the ends of the object at times that are considered simultaneous by an observer moving with the ruler (or moving with the object, in which case the effect is in the opposite direction).

Not true.

4. Jul 2, 2014

### stevendaryl

Staff Emeritus
Relativistic time dilation for an inertial clock is a special case of the Lorentz transforms, and Lorentz contraction is a special case of the Lorentz transforms.

If you have two events $e_1$ and $e_2$, let $\delta x$ be $x_2 - x_1$ and let $\delta t = t_2 - t_1$, as measured in one frame, F. Let $\delta x'$ and $\delta t'$ be the corresponding quantities in another frame, F'. Then the LT says that:

$\delta x' = \gamma (\delta x - v \delta t)$
$\delta t' = \gamma (\delta t - v/c^2 \delta x)$

where $v$ is the velocity of F' relative to F.

So now let's look at some special cases. Suppose that you have a clock that ticks once per second in frame F'. Then letting the two events be two successive ticks of the clock, we have:

$\delta t' = 1$
Because in F', the ticks are one second apart.

$\delta x' = 0$
Because in F', the ticks are at the same location.

Plugging these into the LT gives:
$\delta x'= 0 = \gamma (\delta x - v \delta t)$

So $\delta x= v \delta t$
$\delta t' = 1 = \gamma (\delta t - v/c^2 \delta x)$
$= \gamma(\delta t - v/c^2 \cdot v \delta t)$
$= \gamma \delta t (1 - v^2/c^2)$
$= 1/\gamma \delta t$

So $\delta t = \gamma$. So the time between ticks in frame F is $\gamma$, which is greater than 1.

Now, another special case is a ruler at rest in frame F' (oriented in the direction of motion). Let $e_1$ be the location of one end of the ruler at one moment, and let $e_2$ be the location of the other end at the same time, according to frame F. Let $L$ be the length of the ruler in its own rest frame, F', and let $\tilde{L}[itex] be its length in frame F. Then we have: 1. [itex]\delta x = \tilde{L}$
2. $\delta t = 0$ (because the two events take place at the same time, in F).
3. $\delta x' = L$

Plugging these into the LT gives:
$\delta x' = L = \gamma (\delta x - v \delta t) = \gamma (\tilde{L} - 0) = \gamma \tilde{L}$

So $\tilde{L}= L/\gamma$

So in frame F, the ruler is shorter by a factor of $\gamma$.

5. Jul 2, 2014

### bcrowell

Staff Emeritus
I don't think this is an accurate verbal characterization of your calculations. For example, the Lorentz transformation transforms the coordinates of a single event, whereas your calculations involve two events. What your calculation shows is that length contraction and time dilation can be *derived from* the Lorentz transformation, which is different.

6. Jul 2, 2014

### stevendaryl

Staff Emeritus
Yes, you're right. Length contraction and time dilation follow from special cases the LT, but are not special cases of them.

7. Jul 2, 2014

### pervect

Staff Emeritus
The Lorentz transformations are the transformations for moving between two reference frames. Proper time is an invariant interval along a worldline (a wordline is a curve in spacetime) which is the same for all observers in all frames, and is equal to the time that elapses for a clock on the worldline. Proper time is not a transformation between two reference frames.

In any (flat) frame, the proper time along a worldline is equal to the integral of $\sqrt{dt^2 - dx^2 / c^2}$. A simple special case occurs when you are in a frame where the worldline is x=constant. In that special dx=0, and the Lorentz interval reduces to $\int dt,$ where t is the time coordinate in that particular frame. This simplification only works when dx=0, if dx is not equal to zero you need the whole expression.

If you specify a worldline in one frame, and apply the Lorentz transformation to re-specify the worldline in another frame, you will find that the proper time between two points on the worldline does not change as a consequence of the Lorentz transformation.

The motiviation for proper time and/or proper distance and/or the Lorentz interval is to express physics in a fashion that is independent of any particular choice of coordinates.

Last edited: Jul 2, 2014